from [1]} (9.5.1).
Note that \(\rho _d=\rho _{\Lambda _d}\)  where symbol \(\Lambda _d\)  is given in the proof of
Lemma REF .
It is easy to check that \(\underline{\theta }(\rho _d)=\rho _{d+1}\)  for the dual pair \((_k,_n)\) ,
and then
\(\deg _q(\rho _d)+k(n-k)&= \frac{1}{24}(d-1)d(d+1)(3d+2)+\frac{1}{2}d(d+1)^2 \\&= \frac{1}{24}d(d+1)(d+2)(3d+5) \\&= \deg _q(\rho _{d+1}).\) 
